Abstract

We report on one-dimensional photonic crystals designed to exhibit a pronounced form birefringence at terahertz frequencies. The crystals can be used as volumetric quasioptical elements for a broad frequency range. Theoretical simulations of the dielectric parameters of these structures are presented as well as measurement results of a polymeric crystal that exhibit a birefringence of 0.25 at 300 GHz. As a potential application, the device is exemplarily used as terahertz wave plate.

©2010 Optical Society of America

1. Introduction

The terahertz (THz) frequency range is nowadays explored at an amazing pace. It is expected that THz technology will profit from birefringent materials and devices in a similar way as this is the case for optical systems. Several materials with terahertz birefringence have been reported [14]. Furthermore, a variety of birefringence based devices have already been demonstrated, including wave plates [5,6], filters [7] and phase shifters [8,9].

Recently, birefringent sub-wavelength photonic structures for optical frequencies were investigated [1012]. These structures base on a surface modification of semiconductor materials by etching small grooves into a wafer. This results in a periodic sequence of air and semiconductor material. The birefringence associated with this structure is often called form birefringence [13]. The resulting refractive indices can be adjusted by varying the dimensions of the grooves. Yet, the drawback of this technique is that a controlled surface etching is usually limited to depths of 50 µm to 100 µm. While these sub-100 µm thick structures can be used to build for instance Bragg filter elements for THz frequencies [13], the small interaction length with the THz wave counteracts a general applicability for THz birefringent devices e.g. for THz wave plates.

In this paper we advance the form birefringence approach for the THz frequency range. We present a volumetric one-dimensional photonic crystal (PC) which is formed by sheets of two different alternating polymer layers. This volume element exhibits a pronounced birefringence determined by the design parameters such as the employed material systems and the filling factor. In general, one-dimensional PCs for the THz frequency range are studied as reflectors and filter elements [1416]. Especially designs based on alternating polymer sheets [17] are a promising approach due to the ease of fabrication and their low prices. Here, we propose the application of polymer based photonic crystals as customized birefringent materials.

The paper is organized as follows. First, we give a short theoretical introduction into effective medium theories which describe isotropic material composites. We show that these composite structures can be employed to construct PCs which exhibit form birefringence. Then we discuss the design issues of the proposed devices. To experimentally verify our deductions, we finally present and discuss measured data of a polymeric sample of a one-dimension PC.

2. Theory

To describe the interaction of a composite material system and a THz wave a quasi-static effective medium theory (EMT) can be utilized if the wavelength of the wave is larger than the dimensions of the individual components [18]. In this case, the wave interacts with the structure in the same way as with a bulk material that exhibits effective optical parameter. Furthermore, a periodical structure of these materials results in an effective birefringence of the system [18].

Here, we investigate a structure as illustrated in the Fig. 1 , consisting of alternating layers of two materials with different refractive indices n1 and n2 and thicknesses L1 and L2. The resulting optical properties of this system can be described analytically in the case of a quasi-static regime analogously to a plate capacitance [18]. The quasi-static refractive indices nTE,0 and nTM,0 for a polarization of the electrical field parallel and perpendicular to the layer, respectively, are given by:

nTE,02=f1n12+f2n22,nTM,02=((f1n12)+(f2n22))1,
where f1 and f2 are the volumetric fractions of the individual components. However, this quasi-static assumption is violated as soon as one period, determined by the sum of the two sheet thicknesses Λ=L1+L2, reaches the THz wavelength. Consequently, a more sophisticated approach is required. Rytov [19] has analyzed this specific problem for the case of wavelengths below but still close to the quasi-static limit. He derived a 2nd order EMT which describes the effective optical material parameters nTE and nTM , as a function of the layer properties and the wavelength of the incident electromagnetic wave λ:
nTE2=nTE,02+13(Λλπf1f2(n12n22))2,
nTM2=nTM,02+13(Λλπf1f2(1n121n22)nTE,0nTM,03)2,
where Λ is the period of the structure and ni is the refractive index of the i-th polymeric sheet. As can be seen from the equation, a resulting positive dispersion is expected due to the term Λ/λ even for the case of nondispersive sheet materials. However, this theoretical model is only valid if the dimensions of the sheets slightly exceed the THz wavelengths. For higher frequencies, scattering and waveguiding effects will occur [19] which limits the frequency window of operation for the device. Thus, the dimensions of the layers have to be customized for the specific wavelength of interest. Moreover, a larger difference in the refractive indices of the utilized sheet materials will increase the resulting birefringence.

 figure: Fig. 1

Fig. 1 The principle shape of the investigated PC. The two electrical field components parallel (E) and perpendicular (E) to the layers experience different resulting refractive indices. The propagation direction is illustrated by the wave vector k.

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4. Measurements

For the experimental validation of the proposed design, we fabricate a polymeric sample consisting of 250µm thick layers of polypropylene (PP) and 150 µm thick layers of a compound of PP loaded with 30% of titania (titanium dioxide). The titania particles have a mean size of less than 100nm. At frequencies around 300 GHz the refractive indices of PP and the compound are 1.5 and 2.4, respectively. The compound itself represents an effective medium. Its effective refractive index can be calculated by EMTs as discussed in [20]. Both materials are basically non-dispersive in the frequency window investigated [20]. The individual thicknesses were chosen for a design frequency around 300 GHz.

While coextrusion of the alternating layers is an option for the fabrication of devices of high quality without air inclusions [22], we build a device based on stacked polymer layers to validate the basic principle. The sheets were laser cut to a size of 5mm x 10mm. Afterwards, the alternating sheets were stacked to form a 10mm high device. To characterize the sample, we employed a conventional THz transmission time domain spectrometer as described in detail e.g. in [3]. To extract the resulting refractive indices from the measured data the algorithm presented in [21] was utilized.

The resulting THz signals are shown in Fig. 2 for the case of an orientation of the sheets parallel (TE) and perpendicular (TM) to the polarization of the electrical THz field. For both orientations the THz waveform of the reference signal is chirped due to the dispersion of the device. Moreover, the THz pulse for the TE case is noticeable delayed compared to the TM case which implies different resulting optical lengths for the two polarizations of the electrical field.

 figure: Fig. 2

Fig. 2 The reference pulse (a) and the pulses propagating through the crystal structure: TM case (b) and TE case (c).

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The extracted refractive indices are shown in Fig. 3 compared with simulation results utilizing Eq. (2) and Eq. (3). The inset of the figure shows the refractive indices of PP and the PP compound. In the simulation we assumed resulting air layers between the polymeric sheets of approximately 50µm caused be the imperfect stacking technique [22]. We estimate the dimension of the air spacing by a comparison of the measured device’s thickness to the sum of the thicknesses of the individual layers.

 figure: Fig. 3

Fig. 3 The effective refractive index of the device for the case of a TE and TM wave compared to the predictions of the 2nd order EMT approach. The inset shows the refractive indices of PP and the PP compound.

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The theoretical simulations agree very well with the measured data below 400 GHz. In this frequency range, a pronounced dispersion occurs as predicted from the model. However, the resulting birefringence is relatively constant and measures 0.25. At higher frequencies, the assumptions of the model are violated since the spatial period of the structure reaches the dimension of the wavelength and scattering effects become dominant. In this frequency range, the measured data deviate from the theory’s prediction. As can been seen from the figure, the deviations are in opposite directions for the TE and TM waves and thus, the resulting birefringence decreases. This is expected since a structure with much larger dimensions than the wavelengths would not act as an effective medium and does not exhibit any form birefringence.

To demonstrate one possible application, we adjusted the crystal under an angle of 45° with respect to the polarization of the THz wave. In this configuration the crystal acts as a wave plate and rotates the polarization of the wave as a function of the frequency. This can be characterized by analyzing the resulting transmittance. The wave plate rotates the polarization of the electrical field by 90° if the relative phase shift between the two field components Δφ=2 πLΔn/λ, where Δn is the birefringence and L the device thickness, reaches a multiple of π. In the case of Δφ= π/2 3π/2,… the devices acts as quarter wave plate [1].

The amplitude transmittance T through a polarizer can be described by the Malus’ law, which is given by:

T(ψ)=cos2(ψ),
where ψ denotes the angle between the polarizer and the polarization of the electrical field.

For an orientation of the polarizer parallel to the incident electrical field of the THz wave, the transmission reaches its maxima. In this case the wave plate rotates the polarization by 180° or 360°, meaning Δφ=2π, 4π,... . Minima occur in the case of a rotation by 90° or 270° that corresponds to Δφ=π, 3π,... . Thus, by considering the transmittance TTM and TTE for the individual field components of the THz wave, me model the resulting frequency depended transmission trough the polarizer by:

T(f)=TTETTMcos2(Δφ2),

For the measurement a wire grid polarization filter was placed in front of the receiver antenna to ensure that only the horizontally polarized electrical field component is detected. Figure 4 shows the measured amplitude transmittance and the theoretical predictions based on Eq. (5). Two distinctive minima around 150 GHz and 375 GHz can be observed. The maximum transmission results at 250 GHz. These results agree with the calculations of the relative phase shift considering the frequency depended amount of birefringence. For frequencies above 380 GHz, the crystal does no longer act as a perfect effective medium since the wavelength reaches the size of the layers. Thus, the transmittance of the wave plate comprises deviations from the theoretical shape e.g. the features around 400 GHz. The reason for that is that the effective birefringence reduces in this frequency range as can be seen in Fig. 3. Nevertheless, the modeled transmission, which is based on Eq. (5), the measured refractive indices and the individual transmittance for the TE and TM waves, is in good agreement to the measured data.

 figure: Fig. 4

Fig. 4 The measured amplitude transmittance of the sample orientated 45° in respect to the polarization of the THz wave compared to calculations based on the Malus’ law. The resulting wave plate characteristics modulates the transmission.

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Due to the absorption of the titania loaded PP the maximum transmittance measures 65% only. However, polymers loaded with additives that exhibit lower absorption coefficients, like sapphire particles, or a combination of polymer and air layers could increase the transparency of future devices.

4. Conclusion

We presented an approach for realizing polymer based one-dimensional photonic crystals with pronounced birefringence. By varying the materials employed and the layer thicknesses, the resulting refractive indices and therefore the resulting birefringence can be adjusted. Potential applications include THz wave plates and frequency filter elements. The resulting dispersion could also be utilized for pulse shaping purposes. While coextruded polymeric stacks could provide even higher efficiencies, the first proof of principle device already illuminates the possibilities of this design approach.

Acknowledgments

We thank the Süddeutsches Kunststoff-Zentrum, Würzburg, as well as Steffen Wietzke for providing the PP compound.

References and links

1. M. Reid and R. Fedosejevs, “Terahertz birefringence and attenuation properties of wood and paper,” Appl. Opt. 45(12), 2766–2772 (2006). [CrossRef]   [PubMed]  

2. C. Jördens, M. Scheller, M. Wichmann, M. Mikulics, K. Wiesauer, and M. Koch, “Terahertz birefringence for orientation analysis,” Appl. Opt. 48(11), 2037–2044 (2009). [CrossRef]   [PubMed]  

3. C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010). [CrossRef]  

4. N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010). [CrossRef]   [PubMed]  

5. C.-F. Hsieh, R.-P. Pan, T.-T. Tang, H.-L. Chen, and C.-L. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31(8), 1112–1114 (2006). [CrossRef]   [PubMed]  

6. J.-B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31(2), 265–267 (2006). [CrossRef]   [PubMed]  

7. C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88(10), 101107 (2006). [CrossRef]  

8. C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003). [CrossRef]  

9. C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2 pi liquid crystal terahertz phase shifter,” Opt. Express 12(12), 2625–2630 (2004). [CrossRef]   [PubMed]  

10. S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, “Giant birefringence in multi-slotted silicon nanophotonic waveguides,” Opt. Express 16(11), 8306–8316 (2008). [CrossRef]   [PubMed]  

11. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007). [CrossRef]   [PubMed]  

12. U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, “Implementation of a graded-index medium by use of subwavelength structures with graded fill factor,” J. Opt. Soc. Am. A 22(4), 724–733 (2005). [CrossRef]  

13. C. Kadlec, F. Kadlec, P. Kužel, K. Blary, and P. Mounaix, “Materials with on-demand refractive indices in the terahertz range,” Opt. Lett. 33(19), 2275–2277 (2008). [CrossRef]   [PubMed]  

14. H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004). [CrossRef]  

15. H. Nĕmec, P. Kuzel, L. Duvillaret, A. Pashkin, M. Dressel, and M. T. Sebastian, “Highly tunable photonic crystal filter for the terahertz range,” Opt. Lett. 30(5), 549–551 (2005). [CrossRef]   [PubMed]  

16. W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Quarter-wavelength multilayer interference filter for terahertz waves,” Opt. Commun. 281(9), 2374–2379 (2008). [CrossRef]  

17. D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002). [CrossRef]  

18. D. K. Hale, “The physical properties of composite materials,” J. Mater. Sci. 11(11), 2105–2141 (1976). [CrossRef]  

19. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

20. M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009). [CrossRef]  

21. M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009). [CrossRef]  

22. R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007). [CrossRef]  

References

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  1. M. Reid and R. Fedosejevs, “Terahertz birefringence and attenuation properties of wood and paper,” Appl. Opt. 45(12), 2766–2772 (2006).
    [Crossref] [PubMed]
  2. C. Jördens, M. Scheller, M. Wichmann, M. Mikulics, K. Wiesauer, and M. Koch, “Terahertz birefringence for orientation analysis,” Appl. Opt. 48(11), 2037–2044 (2009).
    [Crossref] [PubMed]
  3. C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
    [Crossref]
  4. N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010).
    [Crossref] [PubMed]
  5. C.-F. Hsieh, R.-P. Pan, T.-T. Tang, H.-L. Chen, and C.-L. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31(8), 1112–1114 (2006).
    [Crossref] [PubMed]
  6. J.-B. Masson and G. Gallot, “Terahertz achromatic quarter-wave plate,” Opt. Lett. 31(2), 265–267 (2006).
    [Crossref] [PubMed]
  7. C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88(10), 101107 (2006).
    [Crossref]
  8. C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003).
    [Crossref]
  9. C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2 pi liquid crystal terahertz phase shifter,” Opt. Express 12(12), 2625–2630 (2004).
    [Crossref] [PubMed]
  10. S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, “Giant birefringence in multi-slotted silicon nanophotonic waveguides,” Opt. Express 16(11), 8306–8316 (2008).
    [Crossref] [PubMed]
  11. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
    [Crossref] [PubMed]
  12. U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, “Implementation of a graded-index medium by use of subwavelength structures with graded fill factor,” J. Opt. Soc. Am. A 22(4), 724–733 (2005).
    [Crossref]
  13. C. Kadlec, F. Kadlec, P. Kužel, K. Blary, and P. Mounaix, “Materials with on-demand refractive indices in the terahertz range,” Opt. Lett. 33(19), 2275–2277 (2008).
    [Crossref] [PubMed]
  14. H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
    [Crossref]
  15. H. Nĕmec, P. Kuzel, L. Duvillaret, A. Pashkin, M. Dressel, and M. T. Sebastian, “Highly tunable photonic crystal filter for the terahertz range,” Opt. Lett. 30(5), 549–551 (2005).
    [Crossref] [PubMed]
  16. W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Quarter-wavelength multilayer interference filter for terahertz waves,” Opt. Commun. 281(9), 2374–2379 (2008).
    [Crossref]
  17. D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
    [Crossref]
  18. D. K. Hale, “The physical properties of composite materials,” J. Mater. Sci. 11(11), 2105–2141 (1976).
    [Crossref]
  19. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  20. M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009).
    [Crossref]
  21. M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
    [Crossref]
  22. R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
    [Crossref]

2010 (2)

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010).
[Crossref] [PubMed]

2009 (3)

C. Jördens, M. Scheller, M. Wichmann, M. Mikulics, K. Wiesauer, and M. Koch, “Terahertz birefringence for orientation analysis,” Appl. Opt. 48(11), 2037–2044 (2009).
[Crossref] [PubMed]

M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009).
[Crossref]

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[Crossref]

2008 (3)

2007 (2)

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
[Crossref]

2006 (4)

2005 (2)

2004 (2)

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2 pi liquid crystal terahertz phase shifter,” Opt. Express 12(12), 2625–2630 (2004).
[Crossref] [PubMed]

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

2003 (1)

C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003).
[Crossref]

2002 (1)

D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
[Crossref]

1976 (1)

D. K. Hale, “The physical properties of composite materials,” J. Mater. Sci. 11(11), 2105–2141 (1976).
[Crossref]

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Abashin, M.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

Abbott, D.

W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Quarter-wavelength multilayer interference filter for terahertz waves,” Opt. Commun. 281(9), 2374–2379 (2008).
[Crossref]

Bandaru, P. R.

Blary, K.

Chen, C.-Y.

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88(10), 101107 (2006).
[Crossref]

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2 pi liquid crystal terahertz phase shifter,” Opt. Express 12(12), 2625–2630 (2004).
[Crossref] [PubMed]

C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003).
[Crossref]

Chen, H.-L.

Cooper, M. L.

Cunningham, J.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

Dobbertin, T.

D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
[Crossref]

Dressel, M.

Duvillaret, L.

H. Nĕmec, P. Kuzel, L. Duvillaret, A. Pashkin, M. Dressel, and M. T. Sebastian, “Highly tunable photonic crystal filter for the terahertz range,” Opt. Lett. 30(5), 549–551 (2005).
[Crossref] [PubMed]

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

Fainman, Y.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, “Implementation of a graded-index medium by use of subwavelength structures with graded fill factor,” J. Opt. Soc. Am. A 22(4), 724–733 (2005).
[Crossref]

Fedosejevs, R.

Fischer, B. M.

W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Quarter-wavelength multilayer interference filter for terahertz waves,” Opt. Commun. 281(9), 2374–2379 (2008).
[Crossref]

Gallot, G.

Garet, F.

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

Hale, D. K.

D. K. Hale, “The physical properties of composite materials,” J. Mater. Sci. 11(11), 2105–2141 (1976).
[Crossref]

Hsieh, C.-F.

Ikeda, K.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

Jansen, C.

N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010).
[Crossref] [PubMed]

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009).
[Crossref]

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[Crossref]

Jördens, C.

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

C. Jördens, M. Scheller, M. Wichmann, M. Mikulics, K. Wiesauer, and M. Koch, “Terahertz birefringence for orientation analysis,” Appl. Opt. 48(11), 2037–2044 (2009).
[Crossref] [PubMed]

Kadlec, C.

Kadlec, F.

Kammoun, A.

D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
[Crossref]

Kim, H.-C.

Knobloch, P.

D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
[Crossref]

Koch, M.

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010).
[Crossref] [PubMed]

C. Jördens, M. Scheller, M. Wichmann, M. Mikulics, K. Wiesauer, and M. Koch, “Terahertz birefringence for orientation analysis,” Appl. Opt. 48(11), 2037–2044 (2009).
[Crossref] [PubMed]

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[Crossref]

M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009).
[Crossref]

R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
[Crossref]

D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
[Crossref]

Krishnamoorthy, A.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

Krumbholz, N.

N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010).
[Crossref] [PubMed]

R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
[Crossref]

Kuzel, P.

H. Nĕmec, P. Kuzel, L. Duvillaret, A. Pashkin, M. Dressel, and M. T. Sebastian, “Highly tunable photonic crystal filter for the terahertz range,” Opt. Lett. 30(5), 549–551 (2005).
[Crossref] [PubMed]

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

Kužel, P.

Levy, U.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, “Implementation of a graded-index medium by use of subwavelength structures with graded fill factor,” J. Opt. Soc. Am. A 22(4), 724–733 (2005).
[Crossref]

Lin, Y.-F.

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88(10), 101107 (2006).
[Crossref]

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2 pi liquid crystal terahertz phase shifter,” Opt. Express 12(12), 2625–2630 (2004).
[Crossref] [PubMed]

Masson, J.-B.

Mikulics, M.

Mittleman, D.

R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
[Crossref]

Mookherjea, S.

Mounaix, P.

Nemec, H.

H. Nĕmec, P. Kuzel, L. Duvillaret, A. Pashkin, M. Dressel, and M. T. Sebastian, “Highly tunable photonic crystal filter for the terahertz range,” Opt. Lett. 30(5), 549–551 (2005).
[Crossref] [PubMed]

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

Nezhad, M.

Pan, C.-L.

C.-F. Hsieh, R.-P. Pan, T.-T. Tang, H.-L. Chen, and C.-L. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31(8), 1112–1114 (2006).
[Crossref] [PubMed]

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88(10), 101107 (2006).
[Crossref]

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2 pi liquid crystal terahertz phase shifter,” Opt. Express 12(12), 2625–2630 (2004).
[Crossref] [PubMed]

C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003).
[Crossref]

Pan, R.-P.

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88(10), 101107 (2006).
[Crossref]

C.-F. Hsieh, R.-P. Pan, T.-T. Tang, H.-L. Chen, and C.-L. Pan, “Voltage-controlled liquid-crystal terahertz phase shifter and quarter-wave plate,” Opt. Lett. 31(8), 1112–1114 (2006).
[Crossref] [PubMed]

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2 pi liquid crystal terahertz phase shifter,” Opt. Express 12(12), 2625–2630 (2004).
[Crossref] [PubMed]

C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003).
[Crossref]

Pang, L.

Pashkin, A.

Rauly, D.

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

Reid, M.

Richard, J.

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

Romeike, D.

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

Rutz, F.

R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
[Crossref]

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Scheller, M.

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010).
[Crossref] [PubMed]

C. Jördens, M. Scheller, M. Wichmann, M. Mikulics, K. Wiesauer, and M. Koch, “Terahertz birefringence for orientation analysis,” Appl. Opt. 48(11), 2037–2044 (2009).
[Crossref] [PubMed]

M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009).
[Crossref]

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[Crossref]

Sebastian, M. T.

Shakfa, M. K.

Tang, T.-T.

Tsai, C.-H.

Tsai, T.-R.

C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003).
[Crossref]

Turchinovich, D.

D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
[Crossref]

Vieweg, N.

Wichmann, M.

Wiesauer, K.

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

C. Jördens, M. Scheller, M. Wichmann, M. Mikulics, K. Wiesauer, and M. Koch, “Terahertz birefringence for orientation analysis,” Appl. Opt. 48(11), 2037–2044 (2009).
[Crossref] [PubMed]

Wietzke, S.

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009).
[Crossref]

Wilk, R.

N. Vieweg, C. Jansen, M. K. Shakfa, M. Scheller, N. Krumbholz, R. Wilk, M. Mikulics, and M. Koch, “Molecular properties of liquid crystals in the terahertz frequency range,” Opt. Express 18(6), 6097–6107 (2010).
[Crossref] [PubMed]

R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
[Crossref]

Withayachumnankul, W.

W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Quarter-wavelength multilayer interference filter for terahertz waves,” Opt. Commun. 281(9), 2374–2379 (2008).
[Crossref]

Xavier, P.

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

Yang, S.-H.

Zentgraf, T.

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88(10), 101107 (2006).
[Crossref]

C.-Y. Chen, T.-R. Tsai, C.-L. Pan, and R.-P. Pan, “Room temperature terahertz phase shifter based on magnetically controlled birefringence in liquid crystals,” Appl. Phys. Lett. 83(22), 4497 (2003).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (1)

D. Turchinovich, A. Kammoun, P. Knobloch, T. Dobbertin, and M. Koch, “Flexible all-plastic mirrors for the THz range,” Appl. Phys., A Mater. Sci. Process. 74(2), 291–293 (2002).
[Crossref]

Compos. Sci. Technol. (1)

C. Jördens, M. Scheller, S. Wietzke, D. Romeike, C. Jansen, T. Zentgraf, K. Wiesauer, and M. Koch, “Terahertz spectroscopy to study the orientation of glass fibres in reinforced plastics,” Compos. Sci. Technol. 70(3), 472–477 (2010).
[Crossref]

J. Appl. Phys. (1)

H. Němec, L. Duvillaret, F. Garet, P. Kuzel, P. Xavier, J. Richard, and D. Rauly, “Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect,” J. Appl. Phys. 96(8), 4072 (2004).
[Crossref]

J. Mater. Sci. (1)

D. K. Hale, “The physical properties of composite materials,” J. Mater. Sci. 11(11), 2105–2141 (1976).
[Crossref]

J. Nanoelectron. Optoelectron. (1)

R. Wilk, N. Krumbholz, F. Rutz, D. Mittleman, and M. Koch, “Dielectric reflectors for terahertz frequencies,” J. Nanoelectron. Optoelectron. 2(1), 77–82 (2007).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. D Appl. Phys. (1)

M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42(6), 065415 (2009).
[Crossref]

Opt. Commun. (2)

M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009).
[Crossref]

W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Quarter-wavelength multilayer interference filter for terahertz waves,” Opt. Commun. 281(9), 2374–2379 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

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Figures (4)

Fig. 1
Fig. 1 The principle shape of the investigated PC. The two electrical field components parallel (E) and perpendicular (E) to the layers experience different resulting refractive indices. The propagation direction is illustrated by the wave vector k.
Fig. 2
Fig. 2 The reference pulse (a) and the pulses propagating through the crystal structure: TM case (b) and TE case (c).
Fig. 3
Fig. 3 The effective refractive index of the device for the case of a TE and TM wave compared to the predictions of the 2nd order EMT approach. The inset shows the refractive indices of PP and the PP compound.
Fig. 4
Fig. 4 The measured amplitude transmittance of the sample orientated 45° in respect to the polarization of the THz wave compared to calculations based on the Malus’ law. The resulting wave plate characteristics modulates the transmission.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

nTE,02=f1n12+f2n22,nTM,02=((f1n12)+(f2n22))1,
nTE2=nTE,02+13(Λλπf1f2(n12n22))2,
nTM2=nTM,02+13(Λλπf1f2(1n121n22)nTE,0nTM,03)2,
T(ψ)=cos2(ψ),
T(f)=TTETTMcos2(Δφ2),

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